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Analysis of the finite element method for transmission/mixed boundary value problems on general polygonal domains
 Electron. Trans. Numer. Anal
"... Abstract. We study theoretical and practical issues arising in the implementation of the Finite Element Method for a strongly elliptic second order equation with jump discontinuities in its coefficients on a polygonal domain Ω that may have cracks or vertices that touch the boundary. We consider in ..."
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Abstract. We study theoretical and practical issues arising in the implementation of the Finite Element Method for a strongly elliptic second order equation with jump discontinuities in its coefficients on a polygonal domain Ω that may have cracks or vertices that touch the boundary. We consider in particular the equation − div(A∇u) = f ∈ H m−1 (Ω) with mixed boundary conditions, where the matrix A has variable, piecewise smooth coefficients. We establish regularity and Fredholm results and, under some additional conditions, we also establish wellposedness in weighted Sobolev spaces. When Neumann boundary conditions are imposed on adjacent sides of the polygonal domain, we obtain the decomposition u = ureg + σ, into a function ureg with better decay at the vertices and a function σ that is locally constant near the vertices, thus proving wellposedness in an augmented space. The theoretical analysis yields interpolation estimates that are then used to construct improved graded meshes recovering the (quasi)optimal rate of convergence for piecewise polynomials of degree m ≥ 1. Several numerical tests are included. Key words. NeumannNeumann vertex, transmission problem, augmented weighted Sobolev space, finite element method, graded mesh, optimal rate of convergence AMS subject classifications. 65N30, 35J25, 46E35, 65N12
AN EQUIVARIANT NONCOMMUTATIVE RESIDUE
, 2006
"... Abstract. Let Γ be a finite group acting on a compact manifold M and A(M) denote the algebra of classical complete symbols on M. We determine all traces on the cross product algebra A(M) ⋊ Γ. These traces appear as residues of certain meromorphic ’zeta ’ functions and can be considered as equivaria ..."
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Abstract. Let Γ be a finite group acting on a compact manifold M and A(M) denote the algebra of classical complete symbols on M. We determine all traces on the cross product algebra A(M) ⋊ Γ. These traces appear as residues of certain meromorphic ’zeta ’ functions and can be considered as equivariant generalization of the noncommutative residue trace. The local formula for these traces depends on more than one component of the complete asymptotic expansion. For instance, the local formula for these traces depends also on derivatives in the normal directions to fixed point manifolds of higher order components. As an application, we obtain a formula for the asymptotic occurrence of an irreducible representation of Γ in the eigenspaces of an invariant positive elliptic operator. We also obtain an new construction for Dixmier trace of an invariant operator. 1.
ANALYSIS OF SCHRÖDINGER OPERATORS WITH INVERSE SQUARE POTENTIALS II: FEM AND APPROXIMATION OF EIGENFUNCTIONS IN THE PERIODIC CASE
"... Abstract. Let V be a periodic potential on R3 that is smooth everywhere except at a discrete set S of points, where it has singularities of the form Z/ρ2, with ρ(x) = x − p  for x close to p and Z is continuous, Z(p)> −1/4 for p ∈ S. We also assume that ρ and Z are smooth outside S and Z is sm ..."
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Abstract. Let V be a periodic potential on R3 that is smooth everywhere except at a discrete set S of points, where it has singularities of the form Z/ρ2, with ρ(x) = x − p  for x close to p and Z is continuous, Z(p)> −1/4 for p ∈ S. We also assume that ρ and Z are smooth outside S and Z is smooth in polar coordinates around each singular point. Let us denote by Λ the periodicity lattice and set T: = R3/Λ. In the first paper of this series [20], we obtained regularity results in weighted Sobolev space for the eigenfunctions of the Schrödingertype operator H = − ∆ + V acting on L2(T), as well as for the induced k–Hamiltonians Hk obtained by resticting the action of H to Bloch waves. In this paper we present two related applications: one to the Finite Element approximation of the solution of (L + Hk)v = f and one to the numerical approximation of the eigenvalues, λ, and eigenfunctions, u, of Hk. We give optimal, higher order convergence results for approximation spaces defined piecewise